PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
MARCELO LACA1 , NADIA S. LARSEN2 , AND SERGEY NESHVEYEV2
Abstract. We develop a general framework for analyzing KMS-states on C∗ -algebras arising from
actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli
and classify its KMS-states for inverse temperatures β 6= 0, 1. In particular, we show that for each
β ∈ (1, 2] there exists a unique KMSβ -state.
Introduction
More than ten years ago Bost and Connes [3] constructed a C∗ -dynamical system with the Galois
group G(Qab /Q) as symmetry group and with phase transition related to properties of zeta and Lfunctions. Since then there have been numerous, and only partially successful, attempts to generalize
the Bost-Connes system to arbitrary number fields, see [5, Section 1.4] for a survey. As was later
emphasized by Connes, the BC-system has yet another remarkable property: there exists a dense
Q-subalgebra such that the maximal abelian extension Qab of Q arises as the set of values of a
ground state of the system on it. If one puts this property as a requirement for an arbitrary number
field, one recognizes that the problem of finding the right analogue of the BC-system is related
to Hilbert’s 12th problem on explicit class field theory. Since the only case (in addition to Q)
for which Hilbert’s problem is completely solved is that of imaginary quadratic fields, these fields
should be the first to investigate. This has been done in recent papers of Connes, Marcolli and
Ramachandran [5, 6, 7, 8]. Connes and Marcolli [5, 6] constructed a GL2 -system, an analogue of the
BC-system with Q∗ replaced by GL2 (Q). Its specialization to a subsystem compatible with complex
multiplication in a given imaginary quadratic field gives the right analogue of the BC-system for
such a field [7, 8]. Later Ha and Paugam [12], inspired by constructions of Connes and Marcolli,
proposed an analogue of the BC-system for an arbitrary number field.
Connes and Marcolli classified KMS-states of the GL2 -system for inverse temperatures β ∈
/ (1, 2].
It is the primary goal of the present paper to elucidate what happens in the critical region (1, 2].
Along the way we develop some general tools for analyzing systems of the type introduced by Connes
and Marcolli, which can be thought of as crossed products of abelian algebras by Hecke algebras.
Our approach to the problem is along the lines of that of the first author in the case of the
BC-system [15]. Namely, in Proposition 3.2 we show that KMS-states correspond to states on the
diagonal subalgebra which are scaled by the action of GL+
2 (Q), or rather by the Hecke operators.
As our first application we recover in Theorem 3.7 the results of Connes and Marcolli. We then
prove our main result, Theorem 4.1, the uniqueness of a KMSβ -state for each β ∈ (1, 2]. The
strategy is similar to that of the third author in the BC-case [18]. Namely, we prove the uniqueness
and ergodicity, under the action of GL+
2 (Q), of the measure defining a symmetric KMSβ -state by
analyzing an explicit formula for the projection onto the space of Mat+
2 (Z)-invariant functions, see
Lemma 4.4 and Corollary 4.7, and then derive from this the main uniqueness result. There are two
Date: September 11, 2006; minor corrections August 27, 2007.
Part of this work was carried out through several visits of the first author to the Department of Mathematics at
the University of Oslo. He would like to thank the department for their hospitality, and the SUPREMA project for
the support.
1
) Supported by the Natural Sciences and Engineering Research Council of Canada.
2
) Supported by the Research Council of Norway.
1
2
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
main complications compared to the BC-case. The first is that instead of semigroup actions we now
have to deal with representations of Hecke algebras. The second is the presence in the system of a
continuous component corresponding to the infinite place. As a result, the critical step now is to
prove the uniqueness of a symmetric, that is, GL2 (Ẑ)-invariant, KMSβ -state, while in the BC-case
the analogous statement is almost obvious. To show this uniqueness we use a deep result of Clozel,
Oh and Ullmo [4] on equidistribution of Hecke points. We point out that, as opposed to the BC-case,
there are many symmetric states for β > 2, which can be easily seen from Theorem 3.7 below.
1. Proper actions and groupoid C∗ -algebras
Let G be a countable group acting on a locally compact second countable space X. The reduced
crossed product C0 (X) or G is the reduced C∗ -algebra of the transformation groupoid G × X with
unit space X, source and range maps (g, x) 7→ x and (g, x) 7→ gx, respectively, and the product
(g, hx)(h, x) = (gh, x).
If the restriction of the action to a subgroup Γ of G is free and proper, we can introduce a new
groupoid Γ\G ×Γ X by taking the quotient of G × X by the action of Γ × Γ defined by
(γ1 , γ2 )(g, x) = (γ1 gγ2−1 , γ2 x).
(1.1)
Thus the unit space of Γ\G ×Γ X is Γ\X, and the product is induced from that on G × X. This
groupoid is Morita equivalent in the sense of [17] to the transformation groupoid G × X. Although
we will not need this result, let us briefly recall the argument. By definition of Morita equivalence
first of all we have to find a space Z with commuting actions of our groupoids. We take Z = G ×Γ X,
the quotient of G × X by the action of Γ given by γ(g, x) = (gγ −1 , γx). The left and right actions
of the groupoid G × X on itself induce a left action of G × X and a right action of Γ\G ×Γ X on Z.
The map Z → Γ\X, Γ(g, x) 7→ Γx, induces a homeomorphism between the quotient of Z by the
action of G × X and the unit space Γ\X of the groupoid Γ\G ×Γ X. Similarly, the map Z → X,
Γ(g, x) 7→ gx, induces a homeomorphism between the quotient of Z by Γ\G ×Γ X and X. Thus the
groupoids are indeed Morita equivariant. Recall then that by [17, Theorem 2.8] the corresponding
reduced C∗ -algebras are Morita equivalent.
If the action of Γ is proper but not free, the quotient space Γ\G×Γ X is no longer a groupoid, since
the composition of classes using representatives will in general depend on the choice of representatives. As was observed in [9] and [5], nevertheless, the same formula for convolution of two functions
as in the groupoid case gives us a well-defined algebra, and by completion we get a C∗ -algebra.
In more detail, consider the space Cc (Γ\G ×Γ X) of continuous compactly supported functions on
Γ\G×Γ X. We consider its elements as (Γ×Γ)-invariant functions on G×X, and define a convolution
of two such functions by
X
(f1 ∗ f2 )(g, x) =
f1 (gh−1 , hx)f2 (h, x).
(1.2)
h∈Γ\G
To see that the convolution is well-defined, assume the support of fi is contained in (Γ×Γ)({gi }×Ui ),
where gi ∈ G and Ui is a compact subset of X. Let {γ1 , . . . , γn } be the set of all elements γ ∈ Γ such
that γg2 U2 ∩ U1 6= ∅. Note that this set is finite since the action of Γ is assumed to be proper. If
f2 (h, x) 6= 0 then there exists γ ∈ Γ such that hγ −1 ∈ Γg2 and γx ∈ U2 . Since the number of γ’s such
that γx ∈ U2 is finite, we already see that the sum above is finite. If furthermore f1 (gh−1 , hx) 6= 0
then replacing h by another representative of the right coset Γh we may assume that gh−1 ∈ Γg1
and hx ∈ U1 . Then if hγ −1 = γ̃g2 with γ̃ ∈ Γ, we get hx = γ̃g2 γx ∈ γ̃g2 U2 . Hence γ̃ = γi for some i,
and therefore g ∈ Γg1 h = Γg1 γi g2 γ. Thus the support of f1 ∗ f2 is contained in the union of the sets
(Γ × Γ)({g1 γi g2 } × U2 ), so f1 ∗ f2 ∈ Cc (Γ\G ×Γ X) and the latter space becomes an algebra. It is
not difficult to check that the convolution is associative.
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
3
Define also an involution on Cc (Γ\G ×Γ X) by
f ∗ (g, x) = f ((g, x)−1 ) = f (g −1 , gx).
(1.3)
If the support of f is contained in (Γ × Γ)({g0 } × U ) for g0 ∈ G and compact U ⊂ X, then the
support of f ∗ is contained in
((Γ × Γ)({g0 } × U ))−1 = (Γ × Γ)({g0 } × U )−1 = (Γ × Γ)({g0−1 } × g0 U ),
so indeed f ∗ ∈ Cc (Γ\G ×Γ X).
For each x ∈ X we define a ∗-representation πx : Cc (Γ\G ×Γ X) → B(`2 (Γ\G)) by
X
f (gh−1 , hx)δΓg ,
πx (f )δΓh =
(1.4)
g∈Γ\G
where δΓg denotes the characteristic function of the coset Γg. It is standard to show that the
operators πx (f ) are bounded, but we include a proof for the reader’s convenience.
Lemma 1.1. For each f ∈ Cc (Γ\G ×Γ X) the operators πx (f ), x ∈ X, are uniformly bounded.
Proof. For ξ1 , ξ2 ∈ `2 (Γ\G) we have
X
|f (gh−1 , hx)| |ξ1 (h)| |ξ2 (g)|
|(πx (f )ξ1 , ξ2 )| ≤
g,h∈Γ\G
1/2 

≤ 
X
|f (gh−1 , hx)| |ξ1 (h)|2 
1/2
X
|f (gh−1 , hx)| |ξ2 (g)|2 

g,h∈Γ\G
.
g,h∈Γ\G
Thus if we denote by kf kI the quantity


X
|f (gh−1 , hx)|,
max
sup
x∈X, h∈G
g∈Γ\G
sup
x∈X, g∈G
X
h∈Γ\G


|f (gh−1 , hx)| ,

we get kπx (f )k ≤ kf kI for any x ∈ X, so it suffices to show that kf kI is finite. Replacing x by h−1 x
and g by gh in the first supremum above, we see that this supremum equals
X
kf kI,s := sup
|f (g, x)|.
x∈X
g∈Γ\G
Observe next that f (gh−1 , hx) = f ∗ (hg −1 , gx), so that the second supremum is equal to kf ∗ kI,s .
Therefore kf kI = max{kf kI,s , kf ∗ kI,s }. It remains to show that kf kI,s is finite for any f ∈
Cc (Γ\G ×Γ X).
Assume the support of f is contained in (Γ × Γ)({g0 } × U ) for some g0 ∈ G and compact U ⊂ X.
Since the action of Γ is proper, there exists n ∈ N such that the sets γi U , i = 1, . . . , n+1, have trivial
intersection for any different γ1 , . . . , γn+1 ∈ Γ. Now if f (g, x) 6= 0 for some g and x, there exists
γ ∈ Γ such that gγ −1 ∈ Γg0 and γx ∈ U . Since the number of γ’s such that γx ∈ U is at most n, we
see that for each x ∈ X the sum in the definition of kf kI,s has at most n nonzero summands. Hence
kf kI,s is finite, and the proof of the lemma is complete.
We denote by Cr∗ (Γ\G ×Γ X) the completion of Cc (Γ\G ×Γ X) in the norm defined by the
representation ⊕x∈X πx , that is,
kf k = sup kπx (f )k.
x∈X
2
` (Γ\G) such
Denoting by Ug the unitary operator on
that Ug δΓh = δΓhg−1 , we get Ug πx (f )Ug∗ =
πgx (f ). Hence kπx (f )k = kπgx (f )k and so the supremum above is actually over G\X.
Using the embedding X ,→ G × X, x 7→ (e, x), we may consider Γ\X as an open subset of
Γ\G ×Γ X, and then the algebra C0 (Γ\X) as a subalgebra of Cr∗ (Γ\G ×Γ X). More generally, any
bounded continuous function on Γ\X defines a multiplier of Cr∗ (Γ\G ×Γ X).
4
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
Lemma 1.2. There exists a conditional expectation E : Cr∗ (Γ\G ×Γ X) → C0 (Γ\X) such that
E(f )(x) = f (e, x) for f ∈ Cc (Γ\G ×Γ X).
Proof. For each x ∈ X define a state ωx on Cr∗ (Γ\G ×Γ X) by
ωx (a) = (πx (a)δΓ , δΓ ).
Then the function E(a) on X defined by E(a)(x) = ωx (a) is bounded by kak. Since E(f )(x) = f (e, x)
for f ∈ Cc (Γ\G ×Γ X), we conclude that E(a) ∈ C0 (Γ\X) for every a ∈ Cr∗ (Γ\G ×Γ X). Thus E is
the required conditional expectation.
Let Y ⊂ X be a Γ-invariant clopen subset. Then, as we already observed, the characteristic
function 1Γ\Y of the set Γ\Y is an element of the multiplier algebra of Cr∗ (Γ\G ×Γ X). Denote by
Γ\G Γ Y the quotient of the space
{(g, x) | g ∈ G, x ∈ Y, gx ∈ Y }
by the action of Γ × Γ defined as in (1.1). Then
1Γ\Y Cc (Γ\G ×Γ X)1Γ\Y = Cc (Γ\G Γ Y ).
Therefore the algebra 1Γ\Y Cr∗ (Γ\G ×Γ X)1Γ\Y , which we shall denote by Cr∗ (Γ\G Γ Y ), is a
completion of the algebra of compactly supported functions on Γ\G Γ Y with convolution product
given by
X
(f1 ∗ f2 )(g, y) =
f1 (gh−1 , hy)f2 (h, y),
h∈Γ\G : hy∈Y
and involution
f ∗ (g, y) = f (g −1 , gy).
Note that πx (1Γ\Y ) is the projection onto the subspace `2 (Γ\Gx ) of `2 (Γ\G), where the subset Gx
of G is defined by
Gx = {g ∈ G | gx ∈ Y },
and then
πx (f )δΓh =
X
f (gh−1 , hx)δΓg
g∈Γ\Gx
for h ∈ Gx and f ∈ Cc (Γ\G Γ Y ). In particular, πx (f ) = 0 if x ∈
/ GY . As we already remarked,
the representations πx and πgx are unitarily equivalent for any g ∈ G. Thus we may conclude that
Cr∗ (Γ\G Γ Y ) is precisely the completion of Cc (Γ\G Γ Y ) in the norm
kf k = sup kπy (f )k.
y∈Y
This is how the algebra Cr∗ (Γ\G Γ Y ) was defined (in a particular case) in [5, Proposition 1.23].
Returning to the algebra Cr∗ (Γ\G ×Γ X), our next goal is to show that under an extra assumption
its multiplier algebra contains other interesting elements in addition to the Γ-invariant functions
on X.
Recall that (G, Γ) is called a Hecke pair if Γ and gΓg −1 are commensurable for any g ∈ G, that is,
Γ ∩ gΓg −1 is a subgroup of Γ of finite index. Equivalently, every double coset of Γ contains finitely
many right (and left) cosets of Γ, so that
RΓ (g) := |Γ\ΓgΓ| < ∞ for any g ∈ G.
Then the space H(G, Γ) of finitely supported functions on Γ\G/Γ is a ∗-algebra with product
X
(f1 ∗ f2 )(g) =
f1 (gh−1 )f2 (h)
h∈Γ\G
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
5
and involution f ∗ (g) = f (g −1 ), see e.g. [13]. This algebra is represented on `2 (Γ\G) by
X
f (gh−1 )δΓg ,
f δΓh =
g∈Γ\G
see [3]. The corresponding completion is called the reduced Hecke C∗ -algebra of (G, Γ) and denoted
by Cr∗ (G, Γ). We shall denote by [g] the characteristic function of the double coset ΓgΓ considered
as an element of the Hecke algebra.
We may consider elements of H(G, Γ) as continuous functions on Γ\G ×Γ X. Although these
functions are not compactly supported in general, the formulas defining the ∗-algebra structure and
the regular representation of H(G, Γ) coincide with (1.2)-(1.4). Furthermore, the convolution of an
element of H(G, Γ) with a compactly supported function on Γ\G ×Γ X gives a compactly supported
function. Indeed, if f1 = [g1 ] and the support of f2 ∈ Cc (Γ\G×Γ X) is contained in (Γ×Γ)({g2 }×U )
for a compact U ⊂ X, then the support of f1 ∗ f2 is contained in (Γ × Γ)(g1 Γg2 × U ). Since Γ\Γg1 Γg2
is finite, we see that f1 ∗ f2 is compactly supported on Γ\G ×Γ X. We may therefore conclude the
following.
Lemma 1.3. If (G, Γ) is a Hecke pair, then the reduced Hecke C∗ -algebra Cr∗ (G, Γ) is contained in
the multiplier algebra of the C∗ -algebra Cr∗ (Γ\G ×Γ X).
It is then tempting to think of Cr∗ (Γ\G ×Γ X) as a crossed product of C0 (Γ\X) by an action of the
Hecke pair (G, Γ). This point of view has been formalized by Tzanev [21] who introduced a notion
of a crossed product of an algebra by an action of a Hecke pair.
Remark 1.4. We defined Cr∗ (Γ\G ×Γ X) assuming that the action of Γ on X is proper. It is however
easy to see that the construction makes sense under the following weaker assumptions: Γ\G ×Γ X
is Hausdorff, and if for a compact set K ⊂ X we put ΓK = {γ ∈ Γ | γK ∩ K 6= ∅} then the set
Γ\ΓgΓK is finite for any g ∈ G. Note that the second assumption is automatically satisfied when
(G, Γ) is a Hecke pair.
2. Dynamics and KMS-states
Assume as above that we have an action of G on X such that the action of Γ ⊂ G is proper, and
Y ⊂ X is a Γ-invariant clopen set. Assume now that we are given a homomorphism
N : G → R∗+ = (0, +∞)
such that Γ is contained in the kernel of N . Then we define a one-parameter group of automorphisms
of Cr∗ (Γ\G ×Γ X) by
σt (f )(g, x) = N (g)it f (g, x) for f ∈ Cc (Γ\G ×Γ X).
More precisely, if we denote by N̄ the selfadjoint operator on `2 (Γ\G) defined by
N̄ δΓg = N (g)δΓg ,
then the dynamics σt is spatially implemented by the unitary operator ⊕x∈X N̄ it on ⊕x∈X `2 (Γ\G).
In other words,
πx (σt (a)) = N̄ it πx (a)N̄ −it for all x ∈ X.
Recall, see e.g. [14], that a semifinite σ-invariant weight ϕ is called a σ-KMSβ -weight if
ϕ(aa∗ ) = ϕ(σiβ/2 (a)∗ σiβ/2 (a))
for any σ-analytic element a. The following result will be the basis of our analysis of KMS-weights.
6
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
Proposition 2.1. Assume the action of G on X is free, so that in particular Γ\G Γ Y is a genuine
groupoid. Then for any β ∈ R there exists a one-to-one correspondence between σ-KMSβ weights ϕ
on Cr∗ (Γ\G Γ Y ) with domain of definition containing Cc (Γ\Y ) and Radon measures µ on Y such
that
µ(gZ) = N (g)−β µ(Z)
(2.1)
for every g ∈ G and every compact subset Z ⊂ Y such that gZ ⊂ Y . Namely, such a measure µ is
Γ-invariant, so it determines a measure ν on Γ\Y such that


Z
Z
X

f (y)dµ(y) =
f (y) dν(t) for f ∈ Cc (Y ),
(2.2)
Y
Γ\Y
y∈p−1 ({t})
where p : Y → Γ\Y is the quotient map, and the associated weight ϕ is given by
Z
E(a)(x)dν(x),
ϕ(a) =
Γ\Y
where E is the conditional expectation defined in Lemma 1.2.
Proof. For Γ = {e} the result is well-known, see e.g. [19, Proposition II.5.4]. For arbitrary Γ the
result can be deduced from the fact that the C∗ -algebra Cr∗ (Γ\G Γ Y ) is Morita equivalent to the
C∗ -algebra 1Y (C0 (X) or G)1Y and general results on KMS-weights on Morita equivalent algebras,
see [16, Theorem 3.2]. However, a more elementary way is to argue as follows.
Since the action of Γ on Y is free, the quotient space Γ\G Γ Y is an etale groupoid. In fact it is
an etale equivalence relation on Γ\Y , or an r-discrete principal groupoid in the terminology of [19].
To see this we have to check that the isotropy group of every point in Γ\Y is trivial, that is, if g ∈ G
is such that gy ∈ Y and p(gy) = p(y) for some y ∈ Y then (g, y) belongs to the (Γ × Γ)-orbit of (e, y).
But if p(gy) = p(y), there exists γ ∈ Γ such that γgy = y. Then γg = e, since the action of G is
free, and therefore (g, y) = (γ −1 , e)(e, y).
It is then standard to show using [19, Proposition II.5.4] that σ-KMSβ weights (with domain of
definition containing Cc (Γ\Y )) on the C∗ -algebra Cr∗ (Γ\G Γ Y ) of the etale equivalence relation are
in one-to-one correspondence with measures ν on Γ\Y with Radon-Nikodym cocycle (p(y), p(gy)) 7→
N (g)β . The latter means the following, see [19, Definition I.3.4]. Assume Y0 is an open subset of Y
such that the map p : Y → Γ\Y is injective on Y0 , and g ∈ G is such that gY0 ⊂ Y . Define an
injective map g̃ : p(Y0 ) → p(gY0 ) by g̃p(y) = p(gy) for y ∈ Y0 , and let g̃∗ ν be the push-forward of the
measure ν under the map g̃, that is, g̃∗ ν(Z) = ν(g̃ −1 (Z)) for Z ⊂ p(gY0 ). Then
dg̃∗ ν
= N (g)β on p(gY0 ).
dν
If we denote by µ the Γ-invariant measure on Y corresponding to ν via (2.2), then to say that the
Radon-Nikodym cocycle of ν is (p(y), p(gy)) 7→ N (g)β is the same as saying that µ satisfies the
scaling condition (2.1).
It will be convenient to extend the measure µ to the set GY .
Lemma 2.2. If µ is a measure on Y as in Proposition 2.1, then it extends uniquely to a Radon
measure on GY ⊂ X satisfying (2.1) for Z ⊂ GY and g ∈ G.
Proof. A more general result on extensions of KMS-weights is proved in [16], but the present particular case has the following elementary proof. Choose Borel subsets Yi ⊂ Y and elements gi ∈ G such
that GY is the disjoint union of the sets gi−1 Yi . There is only one choice for a measure extending µ
and satisfying (2.1) on GY , namely, for a Borel subset Z ⊂ GY let
X
µ(Z) =
N (gi )β µ(gi Z ∩ Yi ).
i
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
7
To show that µ(Z) is independent of any choices and that the extension satisfies (2.1), assume GY
is a disjoint union of sets h−1
j Zj for some hj ∈ G and Borel Zj ⊂ Y . Let g ∈ G. Then
X
X
X
N (gi )β µ(gi gZ ∩ Yi ) =
N (gi )β
µ(gi gZ ∩ Yi ∩ gi gh−1
j Zj )
i
=
i
j
X
X
N (gi )β
i
−β
N (gi gh−1
µ(hj Z ∩ hj g −1 gi−1 Yi ∩ Zj )
j )
j
= N (g)−β
X
= N (g)−β
X
N (hj )β
X
j
µ(hj Z ∩ hj g −1 gi−1 Yi ∩ Zj )
i
N (hj )β µ(hj Z ∩ Zj ).
j
Taking g = e we see that the extension of µ to GY is well-defined. But then for arbitrary g the
above identity reads as µ(gZ) = N (g)−β µ(Z).
Remark 2.3. In the notation of Proposition 2.1 choose a µ-measurable subset U of Y such that
p : Y → Γ\Y is injective on U and p(U ) = Γ\Y . Then the map p induces an isomorphism between
the restriction RG,U of the G-orbit equivalence relation on X to U and the principal groupoid
Γ\G Γ Y . Hence πϕ (Cr∗ (Γ\G Γ Y ))00 is isomorphic to the von Neumann algebra W ∗ (RG,U , µ)
of (RG,U , µ), see [11]. Extend the measure µ to a G-quasi-invariant measure on GY , which we
still denote by µ. Then W ∗ (RG,U , µ) is the reduction of the von Neumann algebra of the G-orbit
equivalence relation on (GY, µ) by the projection 1U . Therefore
πϕ (Cr∗ (Γ\G Γ Y ))00 ∼
= 1U (L∞ (GY, µ) o G)1U .
In some cases an argument similar to the proof of Lemma 2.2 allows us to describe all measures
satisfying (2.1).
Lemma 2.4. Let Y0 be a Γ-invariant Borel subset of Y such that
(i) if gY0 ∩ Y0 6= ∅ for some g ∈ G then g ∈ Γ;
(ii) for any y ∈ Y there exists g ∈ G such that gy ∈ Y0 .
Then any Γ-invariant Borel measure on Y0 extends uniquely to a Borel measure on Y satisfying (2.1).
Proof. Let µ0 be a Γ-invariant measure on Y0 . Since the assumptions imply that Y is a disjoint
union of translates of Y0 by representatives of the right cosets of Γ, that is, Y = th∈Γ\G (h−1 Y0 ∩ Y ),
there is only one choice for a measure µ extending µ0 and satisfying (2.1), namely,
X
µ(Z) =
N (h)β µ0 (hZ ∩ Y0 ).
h∈Γ\G
Since µ0 is Γ-invariant, µ(Z) is independent of the choice of representatives, so all we need to check
is that (2.1) holds. Let g ∈ G. Then
X
X
µ(gZ) =
N (h)β µ0 (hgZ ∩ Y0 ) = N (g)−β
N (hg)β µ0 (hgZ ∩ Y0 ) = N (g)−β µ(Z),
h∈Γ\G
and the proof is complete.
h∈Γ\G
Although the condition for a measure ν on Γ\Y to define a KMS-weight is easier to formulate
in terms of the corresponding Γ-invariant measure on Y , it will also be important to work directly
with ν. For this we introduce the following operators on functions on Γ\X. We shall often consider
functions on Γ\X as Γ-invariant functions on X.
8
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
Definition 2.5. Let G act on a set X and suppose (G, Γ) is a Hecke a pair. The Hecke operator
associated to g ∈ G is the operator Tg on Γ-invariant functions on X defined by
X
1
f (hx).
(Tg f )(x) =
RΓ (g)
h∈Γ\ΓgΓ
Clearly Tg f is again Γ-invariant. It is not difficult to check that the map [g −1 ] → RΓ (g)Tg is a
representation of the Hecke algebra H(G, Γ) on the space of Γ-invariant functions (notice that for
X = G this is exactly the way we defined the regular representation of H(G, Γ), so by decomposing
an arbitrary X into G-orbits one can obtain the general case without any computations).
The following three lemmas will be our main computational tools.
Lemma 2.6. Suppose µ is as in Proposition 2.1 and ν is the measure on Γ\Y determined by (2.2).
Assume further that Y = X, the action of G on X is free and that (G, Γ) is a Hecke pair with
modular function ∆Γ (g) := RΓ (g −1 )/RΓ (g). Then for any positive measurable function f on Γ\X
and g ∈ G we have
Z
Z
Tg f dν = ∆Γ (g)N (g)β
f dν.
Γ\X
Γ\X
Proof. Fix a point x ∈ X. We claim that there exists a neighbourhood U of x such that the sets hU
are disjoint for different h in Γg −1 Γ. Indeed, choose representatives h1 , . . . , hn of the right Γ-cosets
contained in Γg −1 Γ. Since the action of Γ is proper, there exists a neighbourhood U of x such that
if hi U ∩ γhj U 6= ∅ for some i, j and γ ∈ Γ then hi x = γhj x. But since the action of G is free, the
latter equality is possible only when hi = γhj , so that i = j and γ = e. Thus hi U ∩ γhj U = ∅ if
i 6= j or γ 6= e. Since Γg −1 Γ = ∪nk=1 Γhk , this proves the claim.
The set Γg −1 ΓU is therefore a disjoint union of the sets hU , h ∈ Γg −1 Γ. So we can write
X
X
1h−1 ΓU = 1Γg−1 ΓU =
1ΓhU ,
h∈Γ\Γg −1 Γ
h∈Γ\ΓgΓ
Denoting by p : X 7→ Γ\X the quotient map, we can rewrite the above in terms of functions on Γ\X
as
X
RΓ (g)Tg (1p(U ) ) = 1p(Γg−1 ΓU ) =
1p(hU ) .
h∈Γ\Γg −1 Γ
It follows that
Z
RΓ (g)
Tg (1p(U ) )dν =
Γ\X
X
h∈Γ\Γg −1 Γ
ν(p(hU )) =
X
µ(hU ) = RΓ (g −1 )N (g)β ν(p(U )).
h∈Γ\Γg −1 Γ
In other words, the identity in the lemma holds for f = 1p(U ) . Since this is true for any x and
sufficiently small neighbourhood U of x, we get the result.
Notice that by applying the above lemma to the characteristic function of X we get the following:
if a group G acts freely on a space X with a G-invariant measure µ, and Γ is an almost normal
subgroup of G (that is, (G, Γ) is a Hecke pair) such that the action of Γ on X is proper and
0 < µ(Γ\X) < ∞, then ∆Γ (g) = 1 for any g ∈ G. The same is true if we assume that the action
of G on (X, µ) is only essentially free.
Lemma 2.7. Suppose µ is as in Proposition 2.1 and ν is the measure on Γ\Y determined by (2.2).
Assume the action of G on X is free and that (G, Γ) is a Hecke pair. Assume further that Y0 is a
Γ-invariant measurable subset of Y such that if gY0 ∩ Y0 6= ∅ for some g ∈ G then g ∈ Γ. Then for
any g ∈ G such that gY0 ⊂ Y , measurable Z ⊂ Γ\Y0 and positive measurable function f on Γ\Y we
have
Z
Z
f dν = N (g)−β RΓ (g)
ΓgZ
Tg f dν,
Z
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
9
where ΓgZ = p(Γgp−1 (Z)) and p : X → Γ\X is the quotient map. In particular, ν(ΓgZ) =
N (g)−β RΓ (g)ν(Z).
Proof. Suppose Z ⊂ Γ\Y0 is measurable, and choose U ⊂ Y0 measurable such that Z = p(U ) and p
is injective on U . For g ∈ G let h1 , . . . , hn be representatives of the right Γ-cosets contained in ΓgΓ.
We claim that the map p is injective on h1 U, . . . , hn U , and the images of these sets are disjoint.
Indeed, assume p(hi x) = p(hj y) for some i, j and x, y ∈ U , so that γhi x = hj y for some γ ∈ Γ. Since
U ⊂ Y0 , our assumption on Y0 implies h−1
j γhi ∈ Γ. But then, since p is injective on U , we get x = y,
and since the action of Γ is free, we conclude that h−1
j γhi = e. It follows that i = j and hi x = hj y,
which proves the claim.
Furthermore, the union of the disjoint sets p(h1 U ), . . . , p(hn U ) is the set ΓgZ = p(Γgp−1 (Z)).
Hence, since N (hi ) = N (g) for i = 1, . . . , n,
Z
Z
n Z
n Z
X
X
−β
−β
Tg f dν.
f dν =
f ◦ p dµ = N (g)
f (p(hi ·))dµ = N (g) RΓ (g)
ΓgZ
i=1
hi U
i=1
Z
U
The last assertion of the lemma follows by taking f = 1ΓgZ and observing that then (Tg f )(z) = 1
for z ∈ Z.
To formulate the next lemma we introduce the following notation.
Definition 2.8. If β ∈ R and S is a subsemigroup of G containing Γ, then we define
X
X
ζS,Γ (β) :=
N (s)−β =
N (s)−β RΓ (s).
s∈Γ\S
s∈Γ\S/Γ
Lemma 2.9. Suppose µ is as in Proposition 2.1 and ν is the measure on Γ\Y determined by (2.2).
Assume that the action of G on X is free and that (G, Γ) is a Hecke pair. Assume further that Y0
is a measurable Γ-invariant subset of Y , and S a subsemigroup of G containing Γ such that
(i) if gY0 ∩ Y0 6= ∅ for some g ∈ G then g ∈ Γ;
(ii) ∪s∈S sY0 is a subset of Y of full measure;
(iii)ζS,Γ (β) < ∞.
Let HS be the subspace of S-invariant functions in L2 (Γ\Y, ν), that is, functions f such that
f (y) = f (sy) for all s ∈ S and a.a. y ∈ Y . Then
Z
2
|f (t)|2 dν(t);
(1) if f ∈ HS then kf k2 = ζS,Γ (β)
(2) the orthogonal projection P :
Γ\Y0
2
L (Γ\Y, dν)
P f |Sy = ζS,Γ (β)−1
X
→ HS is given by
N (s)−β RΓ (s)(Ts f )(y) for y ∈ Y0 .
(2.3)
s∈Γ\S/Γ
Proof. By condition (i) the sets ΓsY0 are disjoint for s in different double cosets of Γ. Since the
union of such sets is the whole space Y (modulo a set of measure zero), by Lemma 2.7 applied to
Z = Γ\Y0 for any f ∈ L2 (Γ\Y, dν) we get
Z
X Z
X
|f |2 dν =
N (s)−β RΓ (s)
Ts (|f |2 )dν.
(2.4)
kf k22 =
s∈Γ\S/Γ ΓsZ
s∈Γ\S/Γ
Γ\Y0
Since Ts (|f |2 ) = |f |2 for f ∈ HS , this gives (1).
Turning to (2), denote by T the operator on L2 (Γ\Y, dν) defined by the asserted formula for P .
To see that it is well-defined, notice first that the summation in the right hand side of (2.3) is finite
for f in the subspace of L2 -functions supported on a finite collection of sets of the form p(sY0 ),
10
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
s ∈ S, which is a dense subspace of L2 (Γ\Y, dν). Thus the function T f is well-defined for f in this
subspace and, putting αs = ζS,Γ (β)−1 N (s)−β RΓ (s) and using (2.4) twice, we get


Z
Z
X

αs Ts (|f |2 ) dν = kf k22 .
kT f k22 = ζS,Γ (β)
|T f |2 dν ≤ ζS,Γ (β)
Γ\Y0
Γ\Y0
s∈Γ\S/Γ
It follows that T extends to a well-defined contraction. Since T f = f for f ∈ HS , and the image
of T is HS , we conclude that T = P .
3. The Connes-Marcolli system
Consider the group G = GL+
2 (Q) of invertible 2 by 2 matrices with rational coefficients and
positive determinant, and its subgroup Γ = SL2 (Z). For a prime number p consider the field Qp of
p-adic numbers and its compact subring Zp of p-adic integers. We denote by Af the space of finite
Q
adeles of Q, that is, the restricted product of the fields of Qp with respect to Zp , and by Ẑ = p Zp
its maximal compact subring. The field Q is a subfield of Qp , so GL+
2 (Q) can be considered as a
subgroup of GL2 (Qp ). In particular, we have an action of GL+
(Q)
on
Mat2 (Qp ) by multiplication
2
on the left. Moreover, by considering the diagonal embedding of Q into Af we get an embedding
+
+
of GL+
2 (Q) into GL2 (Af ), and thus an action of GL2 (Q) on Mat2 (Af ). In addition GL2 (Q) acts
+
by Möbius transformations on the
upper
halfplane H. Therefore we have an action of GL2 (Q) on
a b
H × Mat2 (Af ) such that for g =
, τ ∈ H and m = (mp )p ∈ Mat2 (Af ),
c d
aτ + b
g(τ, (mp )p ) =
, (gmp )p .
cτ + d
Note that the action of SL2 (Z) is proper, since already the action of SL2 (Z) on H is proper.
The GL2 -system of Connes and Marcolli is now defined as follows, see [5, Section 1.8].
Definition 3.1. The Connes-Marcolli algebra is the C∗ -algebra A = Cr∗ (Γ\G Γ Y ), where G =
GL+
2 (Q), Γ = SL2 (Z), G acts diagonally on X = H × Mat2 (Af ), and Y = H × Mat2 (Ẑ). The
∗
dynamics σ on A is defined by the homomorphism N : GL+
2 (Q) → R+ , N (g) = det(g).
Notice that since Γ\H is not compact, the algebra A is nonunital.
By [5, Lemma 1.28] the action of GL+
2 (Q) on X \ (H × {0}) is free. Recall briefly the reason. If
gm = m for some prime number p and nonzero m ∈ Mat2 (Qp ) then the spectrum of the matrix g
contains 1, and hence g is conjugate in GL+
2 (Q) to an upper-triangular matrix. But then g has
no fixed points in H. Note that what we have actually shown is that the action of GL+
2 (Q) on
H × Mat2 (Qp )× , where Mat2 (Qp )× = Mat2 (Qp ) \ {0}, is free for any prime number p.
Although the action of GL+
2 (Q) on H × {0} is not free, this set can be ignored in the analysis of
KMSβ -states for β 6= 0, see the proof of [5, Proposition 1.30]. Again, recall briefly what happens.
Consider the action of G on X̃ = X \ (H × {0}), put Ỹ = Y \ (H × {0}) ⊂ X̃, and then define
I = Cr∗ (Γ\G Γ Ỹ ). Then I can be considered as an ideal in A, and the quotient algebra A/I is
isomorphic to Cr∗ (Γ\G ×Γ H). Now if ϕ is a σ-KMSβ state on A, the restriction ϕ|I canonically
extends to a KMS-functional on the multiplier algebra of I. Thus we get a KMS-functional ϕ̃ ≤ ϕ
on A. If ϕ̃ 6= ϕ then ϕ − ϕ̃ is a positive nonzero KMS-functional on A which vanishes on I. Hence we
get a KMS-state on A/I ∼
= Cr∗ (Γ\G ×Γ H). By Lemma 1.3 the multiplier algebra of Cr∗ (Γ\G ×Γ H)
contains the reduced Hecke C∗ -algebra Cr∗ (G, Γ). The latter algebra contains in turn the C∗ -algebra
of Z(G)/(Z(G) ∩ Γ), where Z(G) is the center of GL+
2 (Q), that is, the group of scalar matrices. But
since the dynamics scales nontrivially some unitaries in this algebra, the algebra can not have any
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
11
KMSβ -states for β 6= 0. This contradiction shows that ϕ = ϕ̃, so that ϕ is completely determined
by ϕ|I .
Since the action of G on X̃ = H × Mat2 (Af )× , where Mat2 (Af )× = Mat2 (Af ) \ {0}, is free, we can
apply Proposition 2.1 and conclude that there is a one-to-one correspondence between KMSβ -weights
on I with domain of definition containing Cc (Γ\Ỹ ) and measures µ on Ỹ = H × Mat2 (Ẑ)× such that
µ(gZ) = det(g)−β µ(Z) if both Z and gZ are subsets of Ỹ . By Lemma 2.2 we can uniquely extend
any such measure to a measure on X̃ = GỸ = H × Mat2 (Af )× such that µ(gZ) = det(g)−β µ(Z)
for Z ⊂ X̃. To get a state on I = Cr∗ (Γ\G Γ Ỹ ) we need the normalization condition µ(Γ\Ỹ ) = 1
(that is, the Γ-invariant measure µ on Ỹ defines a probability measure on Γ\Ỹ ). Note also that
if β 6= 0 and we have a measure on X = H × Mat2 (Af ) with the same properties as above, then
H × Mat2 (Af )× is a subset of full measure, since scalar matrices act trivially on H and so H cannot
support a measure scaled nontrivially by them.
Summarizing the above discussion we get the following.
Proposition 3.2. For β 6= 0 there is a one-to-one correspondence between σ-KMSβ -states on the
Connes-Marcolli system and Γ-invariant measures µ on H × Mat2 (Af ) such that
µ(Γ\(H × Mat2 (Ẑ))) = 1 and µ(gZ) = det(g)−β µ(Z)
for any g ∈ GL+
2 (Q) and compact Z ⊂ H × Mat2 (Af ).
Denote by Mati2 (Af ) the set of matrices m = (mp )p ∈ Mat2 (Af ) such that det(mp ) 6= 0 for every
prime p. Notice that Mati2 (Af ) is the set of non zero-divisors in Mat2 (Af ). Our next goal is to show
that if β 6= 0, 1 then H×Mati2 (Af ) is a subset of full measure for any measure µ as in Proposition 3.2.
First let us recall the following simple properties of the Hecke pair (G, Γ) = (GL+
2 (Q), SL2 (Z)).
+
(Q)
∩
Mat
(Z).
(Z)
=
GL
Put Mat+
2
2
2
a 0
+
Lemma 3.3. Every double coset of Γ in Mat2 (Z) has a unique representative of the form
0 d
with a, d ∈ N and a|d. Furthermore,
Y
d
a 0
RΓ
=
(1 + p−1 ),
0 d
a
p prime : pa|d
a 0
and as representatives of the right cosets of Γ contained in Γ
Γ we can take the matrices
0 d
ak am
0 al
with k, l ∈ N and m ∈ Z such that kl = d/a, 0 ≤ m < l and gcd(k, l, m) = 1.
In particular, RΓ (g) = RΓ (g −1 ) for every g ∈ GL+
2 (Q).
Proof. See e.g. [13, Chapter IV].
−1
GL+
2 (Z[p ])
Observe
that
if g ∈ Gp then det(g) is a power
p 0
of p, and if we multiply g by a sufficiently large power of
, we get an element in Mat+
2 (Z)
0 p
with determinant a power of p. But
Lemma
3.3 the double coset of Γ containing such an element
by
pk 0
has a representative of the form
, 0 ≤ k ≤ l. We may therefore conclude that Gp is the
0 pl 1 0
+
subgroup of GL2 (Q) generated by Γ and
. Using that a positive rational number is a power
0 p
of p if and only if it belongs to the group of units Z∗q of the ring Zq for all primes q 6= p, we may also
conclude that g ∈ GL+
2 (Q) belongs to Gp if and only if it belongs to GL2 (Zq ) for all q 6= p.
For a prime p put Gp =
⊂
GL+
2 (Q).
12
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
Lemma 3.4. We have GL2 (Qp ) = Gp GL2 (Zp ).
Proof. Let r ∈ GL2 (Qp ). Then rZ2p is a Zp -lattice in Q2p , that is, an open compact Zp -submodule.
By [22, Theorem V.2] there exists a subgroup L ∼
= Z2 of Q2 such that the closure of L in Q2p coincides
2
with rZ2p , and the closure of L in Q2q is Z2q for q 6= p. Choose g ∈ GL+
2 (Q) such that gZ = L. Since
gZ2p = rZ2p , we have g −1 r ∈ GL2 (Zp ). Since gZ2q = Z2q for q 6= p, we also have g ∈ GL2 (Zq ). Hence
g ∈ Gp .
It is also possible to give an elementary proof of Lemma 3.4 using matrix factorization and density
of Z[p−1 ] in Qp .
Lemma 3.5. Let p be a prime and µp a Γ-invariant measure on H × Mat2 (Qp ) such that
µp (H × {0}) = 0, µp (Γ\(H × Mat2 (Zp ))) < ∞ and µp (gZ) = det(g)−β µp (Z)
for g ∈ Gp and Z ⊂ H × Mat2 (Qp ). If β 6= 1, then the set H × GL2 (Qp ) is a subset of full measure
in H × Mat2 (Qp ).
Proof. Denote by ν̃ the measure on Γ\(H × Mat2 (Qp )) defined by the Γ-invariant measure µp . For
a Γ-invariant subset Z of Mat2 (Qp ), the set H × Z is Γ-invariant. We can thus define a measure ν
on the σ-algebra of Γ-invariant Borel subsets of Mat2 (Qp ) by ν(Z) = ν̃(Γ\(H × Z)). Note that since
the action of Γ on Mat2 (Qp ) is not proper and, accordingly, the quotient space Γ\ Mat2 (Qp ) is quite
bad, we do not want to consider Γ-invariant subsets of Mat2 (Qp ) as subsets of this quotient space,
and do not try to define a measure on all Borel subsets of Mat2 (Qp ) out of ν.
If g ∈ Gp and f is a positive Borel Γ-invariant function on Mat2 (Qp ) then by Lemma 2.6 applied
to the function F : (τ, m) 7→ f (m) on Γ\(H × Mat2 (Qp )) we conclude that
Z
Z
Tg f dν =
Tg F dν̃
Mat2 (Qp )
Γ\(H×Mat2 (Qp ))
Z
Z
(3.1)
β
β
= det(g)
F dν̃ = det(g)
f dν.
Γ\(H×Mat2 (Qp ))
Mat2 (Qp )
By assumption we also have ν(Mat2 (Zp )) < ∞. We have to show that the measure of the set of
nonzero singular matrices is zero.
We claim that the set of nonzero singular matrices with coefficients in Qp is the disjoint union of
the sets
0 0
Zk = SL2 (Zp )
GL2 (Zp ), k ∈ Z.
0 pk
This is proved in a standard way: given a nonzero singular matrix we use multiplication by elements
of GL2 (Zp ) on the right to get a matrix with zero first column, and then multiplication by elements
of SL2 (Zp ) on the left to get the required form. To show that the sets do not intersect, observe
that the maximum of the p-adic valuations of the coefficients of a matrix does not change under
multiplication by elements of GL2 (Zp ) on either side.
1 0
Consider the functions fk = 1Zk , k ∈ Z. For g =
we claim that
0 p−1
Tg f0 =
p
1
f0 +
f1 .
p+1
p+1
Indeed, since the action of Gp commutes with the right action of GL2 (Zp ), the function Tg f0 is
GL2 (Zp )-invariant. On the other hand, the sets Zk are clopen subsets of the set of singular matrices,
so that the function f0 is continuous on this set. But then Tg f0 is also continuous. Since Tg f0
is Γ-invariant, and Γ is dense in SL2 (Zp ) (see e.g. [20, Lemma 1.38] for an elementary proof of a
stronger result: Γ is dense in SL2 (Ẑ)), we conclude that Tg f0 is left SL2 (Zp )-invariant. Hence Tg f0
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
13
is
Zk . So toprove the above identity it suffices to check it on the matrices
constant
on the sets
−1
0 0
p
0
p 0
. Since g =
, by Lemma 3.3 we can take the matrices
0 1
0 p−1
0 pk
−1
1 0
p
np−1
,
, 0 ≤ n ≤ p − 1,
0 p−1
0
1
as representatives of the right cosets of Γ contained in ΓgΓ. Then
p−1
1 X
1
0 0
0
0
0 npk−1
(Tg f0 )
=
+
f
.
f
0
0
0 pk
0 pk−1
0
pk
p+1
p+1
n=0
k−1
0
0
0 np
Since the matrices
and
, 1 ≤ n ≤ p − 1, belong to Zk−1 , we see that
0 pk−1
0
pk
1
p
, Tg f0 |Z0 =
, Tg f0 |Zk = 0 for k 6= 0, 1,
p+1
p+1
and this is exactly what we claimed.
It follows from (3.1) that
1
p
p−β ν(Z0 ) =
ν(Z0 ) +
ν(Z1 ).
p+1
p+1
−1
p
0
On the other hand, for g =
we get Tg fk = fk+1 , so that
0 p−1
Tg f0 |Z1 =
p−2β ν(Zk ) = ν(Zk+1 ).
If ν(Z0 ) 6= 0 this implies that p−β is a solution of the quadratic equation
(p + 1)x = 1 + p x2 ,
Thus either p−β = p−1 or p−β = 1. Since β 6= 1 we get β = 0. But then ν(Zk ) = ν(Z0 ) for any k,
and this contradicts ν(Mat2 (Zp )) < ∞. The contradiction shows that ν(Z0 ) = 0. Hence ν(Zk ) = 0
for any k, and we conclude that the measure of the set of singular matrices is zero.
We are now ready to show that for β 6= 0, 1 the set Mat2 (Af ) \ Mati2 (Af ) of zero-divisors has
measure zero.
Corollary 3.6. Assume β 6= 0, 1 and µ is a measure with properties as in Proposition 3.2. Then
H × Mati2 (Af ) is a subset of full measure in H × Mat2 (Af ).
Proof. Fix a prime p. First of all note that the set
{(τ, m) ∈ H × Mat2 (Af ) | mp = 0}
has measure zero. Indeed, as we already remarked before Proposition 3.2, the set H × {0} has
measure zero. So if our claim is not true, the set
{(τ, m) ∈ H × Mat2 (Ẑ)× | mp = 0}
has positive measure. Since the action of Γ on this set is free, there
is a subset U of positive measure
p 0
such that γU ∩ U = ∅ for γ ∈ Γ, γ 6= e. Then for g =
the set Uk = g k U , k ∈ Z, still has
0 p
the property that γUk ∩ Uk = ∅ for γ ∈ Γ, γ 6= e, since g commutes with Γ. As Uk is contained in
H × Mat2 (Ẑ), it follows that µ(Uk ) ≤ 1. On the other hand, µ(Uk ) = p−2βk µ(U ). Letting k → −∞
if β > 0 and k → +∞ if β < 0, we get a contradiction.
Consider now the restriction of µ to the set
Y
H × Mat2 (Qp ) ×
Mat2 (Zq ),
q6=p
14
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
and use the projection onto the first two factors to get a measure µp on H × Mat2 (Qp ). By the first
part of the proof the set H × {0} has µp -measure zero. Since the image of Gp in GL2 (Qq ) lies in
GL2 (Zq ) for q 6= p, the scaling property of µ implies that
µp (gZ) = det(g)−β µp (Z) for Z ⊂ H × Mat2 (Qp ), g ∈ Gp .
Since the action of Γ on H × Mat2 (Qp )× is free, the normalization condition on µ implies that
µp (Γ\(H × Mat2 (Zp )) = 1. Thus µp satisfies the assumptions of Lemma 3.5. Hence H × GL2 (Qp ) is
a set of full µp -measure. This means that the set of points (τ, m) ∈ H × Mat2 (Ẑ) with det(mp ) = 0
has µ-measure zero. By taking the union of such sets for all primes p and multiplying it by elements
i
of GL+
2 (Q) we get a set of measure zero, which is the complement of the set H × Mat2 (Af ).
To get further properties of a measure µ as above, let us recall the following well-known computation, see e.g. [20, Section 3.2] for more general results on formal Dirichlet series. Denote by Sp the
+
semigroup Gp ∩ Mat+
2 (Z). Alternatively, Sp is the set of elements m ∈ Mat2 (Z) with determinant a
nonnegative power of p. Then from Lemma 3.3 we know that as representatives of the right cosets
pk m
of Γ in Sp we can take the matrices
, k, l ≥ 0, 0 ≤ m < pl . Therefore
0 pl
(
∞
X
X
+∞,
if β ≤ 1,
ζSp ,Γ (β) =
det(s)−β =
p−β(k+l) pl =
(3.2)
−β
−1
−β+1
−1
(1 − p ) (1 − p
) , if β > 1.
k,l=0
s∈Γ\Sp
Since Γ = Gp ∩ GL2 (Zp ), we can apply Lemma 2.7 to the group Gp acting on H × Mat2 (Af )× and
the set
Y
Y0 = H × GL2 (Zp ) ×
Mat2 (Zq ).
q6=p
Then for any s ∈ Sp we get
µ(Γ\ΓsY0 ) = det(s)−β RΓ (s)µ(Γ\Y0 ).
The sets ΓsY0 are disjoint for s in different double cosets of Γ, and their union is the set
Y
H × Mati2 (Zp ) ×
Mat2 (Zq ),
q6=p
Mati2 (Zp )
where
= Mat2 (Zp ) ∩ GL2 (Qp ). By Corollary 3.6 the above set is a subset of H × Mat2 (Ẑ)
of full measure for β 6= 0, 1. Therefore we obtain
X
X
1=
µ(Γ\ΓsY0 ) =
det(s)−β RΓ (s)µ(Γ\Y0 ) = ζSp ,Γ (β)µ(Γ\Y0 ).
(3.3)
s∈Γ\Sp /Γ
s∈Γ\Sp /Γ
This gives a contradiction if β < 1. Thus for β < 1, β 6= 0, there are no KMSβ -states. On the other
hand, for β > 1 we get
µ(Γ\Y0 ) = ζSp ,Γ (β)−1 = (1 − p−β )(1 − p−β+1 ).
Assuming now that β > 1 we can perform a similar computation for any finite set of primes
instead of just one prime. Given a finite set F of primes consider the group GF generated by Gp for
+
all p ∈ F . Put also SF = Mat+
2 (Z) ∩ GF . Then SF is the set of matrices m ∈ Mat2 (Z) such that all
prime divisors of det(m) belong to F . Let

 

Y
Y
YF = H × 
GL2 (Zp ) × 
Mat2 (Zq ) .
p∈F
q ∈F
/
Then a computation similar to (3.2) and (3.3) yields
Y
Y
ζSF ,Γ (β) =
(1 − p−β )−1 (1 − p−β+1 )−1 and µ(Γ\YF ) =
(1 − p−β )(1 − p−β+1 ).
p∈F
p∈F
(3.4)
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
15
The intersection of the sets YF over all finite subsets F of prime numbers is the set H × GL2 (Ẑ). So
for β > 2 we get
Y
µ(Γ\(H × GL2 (Ẑ))) =
(1 − p−β )(1 − p−β+1 ) = ζ(β)−1 ζ(β − 1)−1 ,
p
where ζ is the Riemann ζ-function. On the other hand, for β ∈ (1, 2] we get µ(Γ\(H × GL2 (Ẑ))) = 0.
Assume now that β > 2. In this case similarly to (3.2) we have
ζMat+ (Z),Γ (β) = ζ(β)ζ(β − 1).
2
So analogously to (3.3) we get
µ(Γ\Mat+
2 (Z)(H × GL2 (Ẑ))) = ζMat+ (Z),Γ (β)µ(Γ\(H × GL2 (Ẑ))) = 1.
2
We thus see that Mat+
2 (Z)(H × GL2 (Ẑ)) is a subset of H × Mat2 (Ẑ) of full measure. Hence
GL+
(Q)(H×GL
(
Ẑ))
is
a subset of H×Mat2 (Af ) of full measure. By Lemma 3.4 the set GL+
2
2
2 (Q)(H×
GL2 (Ẑ)) is nothing but H × GL2 (Af ).
To summarize, we have shown that for β > 2 the problem of finding all measures µ on H×Mat2 (Af )
satisfying the conditions in Proposition 3.2 reduces to finding all measures on H × GL2 (Af ) such
that
µ(gZ) = det(g)−β µ(Z) and µ(Γ\(H × GL2 (Ẑ))) = ζ(β)−1 ζ(β − 1)−1 .
By Lemma 2.4 any Γ-invariant measure on H×GL2 (Ẑ) extends uniquely to a measure on H×GL2 (Af )
satisfying the scaling condition. Thus we get a one-to-one correspondence between measures µ with
properties as in Proposition 3.2 and measures on Γ\(H × GL2 (Ẑ)) of total mass ζ(β)−1 ζ(β − 1)−1 .
Clearly, extremal measures µ correspond to point masses.
We have thus recovered the following result of Connes and Marcolli [5, Theorem 1.26 and Corollary 1.32].
Theorem 3.7. For the Connes-Marcolli GL2 -system we have:
(i) for β ∈ (−∞, 0) ∪ (0, 1) there are no KMSβ -states;
(ii) for β > 2 there is a one-to-one affine correspondence between KMSβ -states and probability
measures on Γ\(H × GL2 (Ẑ)); in particular, extremal KMSβ -states are in bijection with Γ-orbits in
H × GL2 (Ẑ).
Remark 3.8. This is not exactly what is stated in [5]. First of all, the cases β = 0, 1 require
considerations with singular matrices, and in these cases we do have KMS-states, see Remark 4.8
below. Secondly, the classification of extremal KMSβ -states for β > 2 in [5, Theorem 1.26] is in terms
of invertible Q-lattices up to scaling. To see that our Theorem 3.7(ii) says the same, recall that the
isomorphism from [5, Equation (1.87)] identifies Γ\(H × GL2 (Ẑ)) with the set of invertible Q-lattices
in C up to scaling, and observe that the state ϕβ,l defined in [5, Theorem 1.26(ii)] associated with
l = (τ, ρ) ∈ H × GL2 (Ẑ) is exactly the KMSβ -state corresponding to the orbit Γ(τ, ρ). Since the
Q-lattice picture will not be used in the remaining part of the paper, we omit the details.
4. Uniqueness of the KMSβ -state in the critical region 1 < β ≤ 2
In this section we analyze the Connes-Marcolli system in the region β ∈ (1, 2].
For each such β let us first construct a KMSβ -state, or equivalently, a measure µβ on H×Mat2 (Af )
satisfying the conditions in Proposition 3.2.
16
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
For each prime number p consider the Haar measure on GL2 (Zp ) normalized such that the total
mass is (1 − p−β )(1 − p−β+1 ). By the same argument as in the proof of Lemma 2.4, this measure
extends to a unique measure µβ,p on GL2 (Qp ) such that
X
| det(g)|−β
µβ,p (Z) =
p µβ,p (gZ ∩ GL2 (Zp ))
g∈GL2 (Zp )\GL2 (Qp )
for compact Z ⊂ GL2 (Qp ), where |a|p denotes the p-adic valuation of a. The measure µβ,p satisfies
µβ,p (gZ) = | det(g)|βp µβ,p (Z) for g ∈ GL2 (Qp ).
Since | det(g)|p = 1 for g ∈ GL2 (Zp ), it is clear that µβ,p is left GL2 (Zp )-invariant. But since the Haar
measure on GL2 (Zp ) is biinvariant, we conclude that µβ,p is also right GL2 (Zp )-invariant. By setting
µβ,p (Z) = µβ,p (Z ∩ GL2 (Qp )) for Borel Z ⊂ Mat2 (Qp ) we extend µβ,p to a measure on Mat2 (Qp ).
Using that Mati2 (Zp ) = Sp GL2 (Zp ), similarly to (3.3) we find
µβ,p (Mat2 (Zp )) = ζSp ,Γ (β)µβ,p (GL2 (Zp )) = 1.
Q
Hence we can define a measure on Mat2 (Af ) by µβ,f = p µβ,p . By construction and Lemma 3.5
this is the unique product-measure such that µβ,f (Mat2 (Ẑ)) = 1 and
!β
Y
µβ,f (Z)
(4.1)
µβ,f (gZr) =
| det(gp )|p
p
for Z ⊂ Mat2 (Af ), g = (gp )p ∈ GL2 (Af ) and r ∈ GL2 (Ẑ). Note that since a Haar measure on the
additive group Mat2 (Af ) is a product-measure satisfying (4.1) with β = 2, we see that µ2,f is a Haar
measure on Mat2 (Af ).
Denote by µ∞ the unique GL+
2 (Q)-invariant measure on H such that µ∞ (Γ\H) = 1.
Now put µβ = 2µ∞ × µβ,f . Then µβ satisfies the conditions in Proposition 3.2, so it corresponds
to a KMSβ -state on the Connes-Marcolli C∗ -algebra. Indeed, the scaling condition is satisfied since
Q
−1 for q ∈ Q∗ . The factor 2 is needed for the normalization condition, since the element
+
p |q|p = q
−1 ∈ Γ acts trivially on H, while µβ,f ({±1}\ Mat2 (Ẑ)) = 1/2.
Note that the construction of µβ makes sense for all β > 1.
We can now formulate our main result.
Theorem 4.1. For each β ∈ (1, 2] the state corresponding to the measure µβ is the unique KMSβ state on the Connes-Marcolli system.
We shall prove a slightly stronger result which may look more natural if one leaves aside the
+
∗
motivation for the Connes-Marcolli system. Namely, we replace H by PGL+
2 (R) = GL2 (R)/R .
+
Recall that PGL2 (R) acts transitively on H, and SO2 (R)/{±1} is the stabilizer of the point i ∈ H,
+
so that H = PGL+
2 (R)/PSO2 (R). Denote by µ̄∞ the Haar measure on PGL2 (R) normalized such
+
+
that µ̄∞ (Γ\PGL2 (R)) = 1. Define then a measure on PGL2 (R) × Mat2 (Af ) by µ̄β = 2µ̄∞ × µβ,f .
Theorem 4.2. For β ∈ (1, 2] the measure µ̄β is the unique Γ-invariant measure on the space
PGL+
2 (R) × Mat2 (Af ) such that
−β
µ̄β (Γ\(PGL+
µ̄β (Z)
2 (R) × Mat2 (Ẑ))) = 1 and µ̄β (gZ) = det(g)
+
for compact Z ⊂ PGL+
2 (R) × Mat2 (Af ) and g ∈ GL2 (Q).
Theorem 4.1 follows from the above theorem since every measure µ on H × Mat2 (Af ) satisfying
the conditions in Proposition 3.2 gives rise to a measure µ̄ on PGL+
2 (R) × Mat2 (Af ) satisfying the
conditions in Theorem 4.2 by the formula
!
Z
Z
Z
f dµ̄ =
f (· g)dg dµ,
PGL+
2 (R)×Mat2 (Af )
H×Mat2 (Af )
PSO2 (R)
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
17
where for x = (h, m) ∈ PGL+
2 (R) × Mat2 (Af ) and g ∈ PSO2 (R) we put xg = (hg, m), and different
measures µ give rise to different µ̄’s.
Turning to the proof of Theorem 4.2 our first goal is to show uniqueness of µ̄β under the additional
assumption of invariance under the right action of GL2 (Ẑ) on Mat2 (Af ).
Let F be a finite set of prime numbers. Recall that we denote by SF the semigroup of matrices
m ∈ Mat+
2 (Z) such that all prime divisors of det(m) belong to F . We then introduce an operator TF
on the space of bounded functions on Γ\PGL+
2 (R) by
X
det(s)−β RΓ (s)(Ts f )(τ ).
(4.2)
(TF f )(τ ) = ζSF ,Γ (β)−1
s∈Γ\SF /Γ
Denote by ν̄∞ the measure on Γ\PGL+
2 (R) defined by µ̄∞ . The following result is a key point in
our argument for uniqueness of the GL2 (Ẑ)-invariant measure.
Lemma 4.3. For any finite set J of prime numbers, f ∈ Cc (Γ\PGL+
2 (R)), ε > 0 and compact
subset Ω ⊂ Γ\PGL+
(R),
there
exists
a
finite
set
F
of
prime
numbers
that
is disjoint from J and
2
satisfies
Z
f dν̄∞ < ε for all τ ∈ Ω.
(TF f )(τ ) −
+
Γ\PGL2 (R)
Proof. By [4, Theorem 1.7] and Remark (3) following it, see also [10] for an alternative proof of a
slightly weaker result, there exists a constant M such that
Z
ε
f dν̄∞ <
(Tg f )(τ ) −
+
2
Γ\PGL2 (R)
for τ ∈ Ω and any g ∈ GL+
2 (Q) with RΓ (g) > M . We may assume that M is such that p < M for
any p ∈ J. Let F be a finite set of prime numbers greater than M . Then from Lemma 3.3 we see
that RΓ (s) > M for any s ∈ SF such that ΓsΓ contains a nonscalar diagonal matrix. On the other
hand,
!
∞
X
Y X
Y
det(s)−β =
p−2βk =
(1 − p−2β )−1 ≤ ζ(2β).
s∈Γ\SF /Γ:
s scalar
p∈F
k=0
p∈F
Since the operators Tg are contractions in the supremum-norm, we can find C > 0 such that
Z
+
(T
f
)(τ
)
−
f
dν̄
g
∞ ≤ C for τ ∈ Ω and g ∈ GL2 (Q).
+
Γ\PGL2 (R)
Therefore by considering separately the summation over double cosets with nonscalar and scalar
representatives we get
Z
ε
ζ(2β)
C for any τ ∈ Ω.
f dν̄∞ ≤ +
(TF f )(τ ) −
+
2 ζSF ,Γ (β)
Γ\PGL2 (R)
Recall that by (3.4)
ζSF ,Γ (β) =
Y
(1 − p−β )−1 (1 − p−β+1 )−1 .
p∈F
Since for β ≤ 2 this product diverges as F increases, we see that by choosing sufficiently large F we
can make the second summand in the estimate above arbitrarily small, hence we are done.
We can now analyze the case of measures on PGL+
2 (R) × Mat2 (Af ) that are invariant under the
right action of GL2 (Ẑ) on the second factor.
18
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
Lemma 4.4. The measure µ̄β is the unique right GL2 (Ẑ)-invariant measure on PGL+
2 (R)×Mat2 (Af )
that satisfies the conditions in Theorem 4.2. Furthermore, the action of GL+
(Q)
on the space
2
+
(PGL2 (R) × (Mat2 (Af )/GL2 (Ẑ)), µ̄β ) is ergodic.
Proof. The measure µ̄β is right GL2 (Ẑ)-invariant and satisfies the conditions in Theorem 4.2 by
construction. Suppose µ̃ is another such measure. Let ν̃ and ν̄β be the measures on the quotient
space Γ\(PGL+
2 (R) × Mat2 (Af )) defined by µ̃ and µ̄β , respectively. Let H be the subspace of
+
Mat2 (Z)-invariant functions in L2 (Γ\(PGL+
2 (R) × Mat2 (Ẑ)), dν̃), and denote by P the orthogonal
projection onto H. Our first goal is to compute how P acts on GL2 (Ẑ)-invariant functions.
Let F be a nonempty finite set of prime numbers. Apply Lemma 2.9(2) to the group GF , the
semigroup SF , the set Y = PGL+
2 (R) × Mat2 (Ẑ) and the subset
Y
Y
Mat2 (Zq )
YF = PGL+
GL2 (Zp ) ×
2 (R) ×
p∈F
q ∈F
/
in place of Y0 . Note that we can do this because SF YF coincides with
Y
Y
i
Mat2 (Zq ),
PGL+
(R)
×
Mat
(Z
)
×
p
2
2
p∈F
q ∈F
/
which by Corollary 3.6 (or rather its analogue with H replaced by PGL+
2 (R)) is a subset of Y of
full measure. Thus, denoting by PF the projection onto the subspace of SF -invariant functions, for
f0 ∈ L2 (Γ\(PGL+
2 (R) × Mat2 (Ẑ)), dν̃) we have
X
PF f0 |SF x = ζSF ,Γ (β)−1
det(s)−β RΓ (s)(Ts f0 )(x) for each x ∈ YF .
(4.3)
s∈Γ\SF /Γ
Given a finite set J of prime numbers which is disjoint from F , and a bounded Borel function f on
Γ\PGL+
2 (R), apply (4.3) to the function f0 = fJ , where fJ is defined by
(
f (τ ), if x = (τ, m) ∈ YJ ,
fJ (x) =
0,
otherwise.
Then using the operator TF defined in (4.2), we can write
PF fJ = (TF f )J .
Assume now that f is continuous and compactly supported. By Lemma
R 4.3 we can find a sequence
{Fn }n of finite sets disjoint from J such that {TFn f }n converges
to f dν̄∞ uniformly
R
R on compact
sets. Hence the sequence {PFn fJ }n converges weakly in L2 to f dν̄∞ (1Γ\PGL+ (R) )J = f dν̄∞ 1Γ\YJ .
2
Since P PF = P for every F , we get
Z
P fJ = f dν̄∞ P 1Γ\YJ .
Using formula (4.3) for the set J instead of F , we also see that PJ 1Γ\YJ is the constant function
ζSJ ,Γ (β)−1 . Using again that P PJ = P , we therefore obtain
Z
−1
P fJ = ζSJ ,Γ (β)
f dν̄∞ .
(4.4)
Γ\PGL+
2 (R)
Since the space H contains nonzero constant functions, this in particular implies that
Z
Z
−1
fJ dν̃ = ζSJ ,Γ (β)
f dν̄∞ ,
Γ\PGL+
2 (R)
so that
R
fJ dν̃ is the same for every µ̃.
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
19
To extend the result to allQGL2 (Ẑ)-invariant functions, fix a finite nonempty set J of prime
numbers, and consider a right p∈J GL2 (Zp )-invariant bounded Borel function f on


Y
Γ\ PGL+
Mat2 (Zp ) .
2 (R) ×
p∈J
We may consider f as a function on Γ\(PGL+
2 (R)×Mat2 (Ẑ)). Then f is right GL2 (Ẑ)-invariant, and
the space spanned by such functions for all J’s is dense in the space of square integrable GL2 (Ẑ)invariant functions. Applying again formula (4.3) for the projection PJ (for J in place of F ), we
see that PJ f is again a function whose value at (τ, m) ∈ PGL+
2 (R) × Mat2 (Ẑ) depends only on τ
and mp with p ∈ J. The formula also shows that PJ commutes with the action of GL2 (Ẑ), so
PJ f is GL2 (Ẑ)-invariant. Since GL2 (Zp ) acts transitively on itself, this shows that the value of PJ f
at (τ, m) with mp ∈ GL2 (Zp ) for p ∈ J depends only on τ . In other words, on the space Γ\YJ
introduced above, the function PJ f is a bounded Borel function of the form f˜J for some function f˜
˜
on Γ\PGL+
2 (R). An important point is that f depends on f but not on µ̃. By Lemma 2.9(1) and
the polarization identity we have
Z
Z
Z
Z
˜
PJ f dν̃ = ζSJ ,Γ (β)
PJ f dν̃ = ζSJ ,Γ (β)
fJ dν̃ =
f˜dν̄∞ .
Γ\YJ
Γ\YJ
Γ\PGL+
2 (R)
R
R
R
Since f dν̃ = PJ f dν̃, we see again that f dν̃ is the same for any µ̃. It therefore follows that
R
R
f dν̃ = f dν̄β for any bounded Borel GL2 (Ẑ)-invariant function on Γ\(PGL+
2 (R) × Mat2 (Ẑ)).
Since ν̃ is GL2 (Ẑ)-invariant by assumption, we have ν̃ = ν̄β and hence µ̃ = µ̄β .
To prove ergodicity assume Z0 is a left GL+
2 (Q)-invariant and right GL2 (Ẑ)-invariant µ̄β -measur+
(R)
×
Mat
(A
)
of
positive
measure. Since GL+
able subset of PGL+
2
f
2 (Q)(PGL2 (R) × Mat2 (Ẑ)) =
2
+
PGL+
2 (R)×Mat2 (Af ), it follows that the set Z0 ∩(PGL2 (R)×Mat2 (Ẑ)) has positive measure. Hence
λ = µ̄β (Γ\(Z0 ∩ (PGL+
2 (R) × Mat2 (Ẑ)))) > 0. It follows that the measure µ̃ defined by
µ̃(Z) = λ−1 µ̄β (Z0 ∩ Z)
is right GL2 (Ẑ)-invariant and satisfies the conditions in Theorem 4.2. Hence µ̃ = µ̄β , and consequently the complement of Z0 has µ̄β -measure zero.
+
We aim to prove that the action of GL+
2 (Q) on (PGL2 (R) × Mat2 (Af ), µ̄β ) is ergodic. The next
step is to consider the action on Mat2 (Af ) alone.
Lemma 4.5. The action of GL+
2 (Q) on (Mat2 (Af ), µβ,f ) is ergodic.
Proof. The proof is similar to that of the previous lemma, but requires a much simpler result than
Lemma 4.3.
Consider the space L2 (Mat2 (Ẑ), dµβ,f ) and the subspace H of Mat+
2 (Z)-invariant functions. It
suffices to show that H consists of constant functions. Denote by P the orthogonal projection
onto H.
For a finite set F of prime numbers denote by PF the projection onto the space of SF -invariant
functions. Put also
Y
Y
YF =
GL2 (Zp ) ×
Mat2 (Zq ).
p∈F
q ∈F
/
Then similarly to (4.3) for any Γ-invariant function f ∈ L2 (Mat2 (Ẑ), dµβ,f ) we have
X
PF f |SF m = ζSF ,Γ (β)−1
det(s)−β RΓ (s)(Ts f )(m) for m ∈ YF .
s∈Γ\SF /Γ
(4.5)
20
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
This can be either proved similarly to Lemma 2.9(2) or deduced from that lemma by identifying the
space of Γ-invariant functions with the subspace of L2 (Γ\(PGL+
2 (R) × Mat2 (Ẑ)), dν̄β ) of functions
depending only on the second coordinate.
Q
For a finite set J of primes disjoint from F , and a left Γ-invariant function f on p∈J GL2 (Zp )
define a function fJ by
(
f ((mp )p∈J ), if mp ∈ GL2 (Zp ) for p ∈ J,
fJ (m) =
0,
otherwise.
Q
Since f is Γ-invariant and Γ is dense in p∈J SL2 (Zp ), f is invariant with respect to multiplication
on the leftQby elements of the latter group. In other words, the value of f at m depends only on
det(m) ∈ p∈J Z∗p . Therefore functions of the form f (m) = χ(det(m)), where χ is a character of the
Q
Q
compact abelian group p∈J Z∗p , span a dense subspace of Γ-invariant functions on p∈J GL2 (Zp ).
But if f = χ ◦ det, we have
(Ts f )(m) = χ(det(m))χ(det(s))
Q
for s ∈ SF and m ∈ p∈J GL2 (Zp ). Applying now (4.5) to the function fJ and using a calculation
similar to (3.2) and (3.4), we get
X
PF fJ |SF m = χ(det((mp )p∈J ))ζSF ,Γ (β)−1
det(s)−β RΓ (s)χ(det(s))
s∈Γ\SF /Γ
= χ(det((mp )p∈J ))
Y
p∈F
(1 − p−β )(1 − p−β+1 )
.
(1 − χ(p)p−β )(1 − χ(p)p−β+1 )
If the character χ is nontrivial, by choosing F large enough the product above can be made arbitrarily
small by elementary properties of Dirichlet series (this was used already for the classification of KMSstates of the Bost-Connes system in [3], see also [18]). Since P PF = P , we conclude that P fJ = 0.
On the other hand, if χ is trivial then fJ = 1YJ . Then applying (4.5) with J in place of F we get
PJ fJ = ζSJ ,Γ (β)−1 . In either case we see that P fJ is constant.
Q
Let now f be a function on p∈J GL2 (Zp ) which is no longer left Γ-invariant. Since Γ is dense
in SL2 (Ẑ), any function in H is SL2 (Ẑ)-invariant. Hence to compute P fJ we can first apply to fJ
the projection Q onto the subspace of SL2 (Ẑ)-invariant functions. But Q is given by averaging over
SL2 (Ẑ)-orbits. We then see that QfJ = f˜J , where
Z
˜
f (m) = Q
f (gm)dg.
p∈J
SL2 (Zp )
Hence P fJ = P QfJ = P f˜J is again a constant function.
To extend the result to all functions on Mat2 (Ẑ), for each s ∈ Mat+
2 (Z) we introduce an operator Vs
2
on the space L (Mat2 (Ẑ), dµβ,f ) by letting (Vs h)(m) = h(sm). Then Vs P = P . Using the scaling
condition we see that det(s)−β/2 Vs is a coisometry with initial space L2 (s Mat2 (Ẑ), dµβ,f ). It follows
that the adjoint operator is given by
(
det(s)β h(s−1 y), if y ∈ s Mat2 (Ẑ),
(Vs∗ h)(y) =
0,
otherwise.
In particular, we see that if s ∈ SJ for some finite set J then both operators Vs and Vs∗ preserve the
space of functions f such that f (m) depends only on mp with p ∈ J. But then if f is such a function
with support on YJ , the function Vs∗ f has support on sYJ . Since Vs P = P , we have P Vs∗ = P and
thus P Vs∗ f = P f is a constant. We thus see that the image of a dense space of functions consists of
constant functions.
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
21
The following simple trick will allow us to combine the two previous lemmas. It expounds a
remark in [18].
Proposition 4.6. Assume we have mutually commuting actions of locally compact second countable
groups G1 , G2 and G3 on a Lebesgue space (X, µ). Suppose that
(i) the actions of G1 on (X/G2 , µ) and (X/G3 , µ) are ergodic;
(ii) G2 is connected and G3 is compact totally disconnected.
Then the action of G1 on (X, µ) is ergodic.
Here by quotient spaces we mean quotients in measure theoretic sense. So by definition
L∞ (X/Gi , µ) = L∞ (X, µ)Gi .
Proof of Proposition 4.6. By assumption the action of G1 × G3 on X is ergodic. In other words,
the action of G3 on X/G1 is ergodic. Since G3 is compact, we can then identify X/G1 with a
homogeneous space of G3 , say G3 /H, where H is a closed subgroup of G3 , see e.g. [23, Section 2.1].
Since G1 × G2 acts ergodically on X, we have an ergodic action of G2 on X/G1 = G3 /H. Since this
action commutes with the action of G3 on G3 /H by left translations, it is given by right translations,
that is, by a measurable homomorphism G2 → N (H)/H, where N (H) is the normalizer of H in G3 .
Such a homomorphism is automatically continuous (see e.g. [23, Theorem B.3]), and since G2 is
connected and N (H)/H is totally disconnected, the homomorphism must be trivial. But since the
action of G2 is ergodic this means that H = G3 , so that X/G1 is a single point. Thus the action
of G1 is ergodic.
+
Corollary 4.7. The left action of GL+
2 (Q) on (PGL2 (R) × Mat2 (Af ), µ̄β ) is ergodic.
Proof. The proof is a straightforward application of Proposition 4.6 with G1 = GL+
2 (Q), G2 =
+
+
PGL2 (R) and G3 = GL2 (Ẑ), so that G1 acts on PGL2 (R) × Mat2 (Af ) by multiplication on the left
and G2 and G3 act by multiplication on the right on the corresponding factor. That the actions
of G1 on the quotient spaces are ergodic is given by Lemma 4.4 and Lemma 4.5.
Proof of Theorem 4.2. We follow an argument similar to that of [3, Theorem 25]. Note first that µ̄β
is right GL2 (Ẑ)-invariant and satisfies the conditions in Theorem 4.2 by construction. Denote by Kβ
the affine set of measures on PGL+
2 (R) × Mat2 (Af ) satisfying the conditions in Theorem 4.2. Let
Cβ be a cone with base Kβ . Denote by v0 its vertex. The cone Cβ has the structure of a Choquet
simplex. Namely, similarly to Proposition 3.2 it can be identified with the set of KMSβ -states on B ∼ ,
where
+
B = Cr∗ (Γ\GL+
2 (Q) Γ (PGL2 (R) × Mat2 (Ẑ))),
B ∼ is obtained from B by adjoining a unit, and v0 corresponds to the state on B ∼ with kernel B.
Denote by ϕ̄ the state corresponding to µ̄β . Then by Remark 2.3 the algebra πϕ̄ (B ∼ )00 is a reduction
of the von Neumann algebra of the orbit equivalence relation defined by the action of GL+
2 (Q) on
+
(PGL2 (R) × Mat2 (Af ), µ̄β ). By Corollary 4.7 this von Neumann algebra is a factor. Hence πϕ̄ (B ∼ )00
is also a factor, and therefore µ̄β is an extremal point of Cβ . The group GL2 (Ẑ) acts on Cβ , and by
virtue of Lemma 4.4 the segment R[µ̄β , v0 ] is the set of GL2 (Ẑ)-invariant points. Suppose now v ∈ Cβ
is an extremal point. Then w = GL2 (Ẑ) gv dg ∈ Cβ where each gv is also an extremal point of Cβ .
But because of its GL2 (Ẑ) invariance, w lies on [µ̄β , v0 ] and hence is also a convex combination of
the extremal points µ̄β and v0 . Since w is the barycenter of a unique probability measure on the set
of extremal points, we conclude that either v = µ̄β or v = v0 . Thus Cβ = [µ̄β , v0 ] and Kβ = {µ̄β }.
This completes the proof of Theorem 4.2.
Remark 4.8. We have classified KMSβ -states of the Connes-Marcolli system for β 6= 0, 1. Let us
now briefly discuss the cases β = 0, 1.
(i) If β = 0 then by Lemma 3.5 and the considerations following Corollary 3.6 one can conclude that
22
M. LACA, N. S. LARSEN, AND S. NESHVEYEV
×
there are no nonzero finite traces on I = Cr∗ (Γ\GL+
2 (Q) Γ (H × Mat2 (Ẑ) )). Therefore the only
∗
KMS0 -states, that is, σ-invariant traces, are those coming from A/I = Cr (Γ\GL+
2 (Q) ×Γ H). There
is a canonical trace defined by the GL+
(Q)-invariant
measure
µ
on
H.
Notice
that though the
∞
2
+
action of GL2 (Q) on H is not free and so Proposition 2.1 is not immediately applicable, the action
∗
of GL+
2 (Q)/Q is free in the measure theoretic sense, and this is enough to check the trace property.
This is probably the unique such trace.
(ii) If β = 1 then, as we know, KMS1 -states still correspond to measures satisfying the scaling
condition. By the first part of the proof of Corollary 3.6 and our considerations following that
corollary, the set of points (τ, m) ∈ H × Mat2 (Af ) with mp 6= 0, det(mp ) = 0 for every p, is a subset
of full measure. Such measures indeed exist. Let µ0f be the Haar measure on the locally compact
group A2f normalized such that µ0f (Ẑ2 ) = 1. We may consider µ0f as a measure on Mat2 (Af ) by
identifying A2f with the set of matrices with zero first column. Then µ0 = 2µ∞ × µ0f is a measure
with the required properties. Using the action of GL2 (Ẑ) by multiplication on the right we can then
construct infinitely many such measures (notice that the stabilizer of µ0 in GL2 (Ẑ) is the group of
upper triangular matrices). We conjecture that this way one gets all extremal KMS1 -states.
Remark 4.9. Let 1 < β ≤ 2, and denote by ϕβ the unique KMSβ -state on the Connes-Marcolli
C∗ -algebra A. It is easy to describe the flow of weights of the factor πϕβ (A)00 . Let us first consider
+
the algebra B = Cr∗ (Γ\GL+
2 (Q) Γ (PGL2 (R) × Mat2 (Ẑ))) and the state ϕ̄β on B corresponding
to µ̄β , and describe the flow of weights of πϕ̄β (B)00 . By Remark 2.3, equivalently we want to describe
the flow of weights of the orbit equivalence relation defined by the ergodic action of GL+
2 (Q) on
(R)
×
Mat
(A
),
µ̄
).
(PGL+
2
f
β
2
The group R∗+ acts on the measure space (R∗+ × PGL+
2 (R) × Mat2 (Af ), λ × µ̄∞ × µβ,f ), where λ
is a measure in the Lebesgue measure class, by
t(s, h, ρ) = (t−1/β s, h, ρ).
The flow of weights is induced by this action on the quotient of the space by the action of GL+
2 (Q)
defined by
g(s, h, ρ) = (det(g)s, gh, gρ).
+
∗
We have an isomorphism GL+
2 (R)/{±1} → R+ × PGL2 (R), g 7→ (det(g), ḡ), where ḡ denotes the
+
∗
class of g in PGL+
2 (R). So instead of the space R+ × PGL2 (R) × Mat2 (Af ) we may consider
+
+
(GL2 (R)/{±1}) × Mat2 (Af ). We may further replace GL2 (R) by GL2 (R), but instead of the action
of GL+
2 (Q) we then have to consider the action of GL2 (Q). Finally, replace GL2 (R) by Mat2 (R),
and so instead of GL2 (R) × Mat2 (Af ) consider Mat2 (A), where A = R × Af is the full adele space.
To summarize, R∗+ acts on Mat2 (A) = Mat2 (R) × Mat2 (Af ) by t(m, ρ) = (t−1/2β m, ρ), and the flow
of weights of the factor πϕ̄β (B)00 is induced by this action on the quotient of the measure space
(Mat2 (R) × Mat2 (Af ), λ∞ × µβ,f ), where λ∞ is the usual Lebesgue measure on Mat2 (R) ∼
= R4 , by
the action of GL2 (Q) × {±1} defined by (g, s)(m, ρ) = (gms, gρ).
Denote the measure λ∞ × µβ,f on Mat2 (A) by λβ . Note that λ2 is a Haar measure on the additive
group Mat2 (A).
Similarly, by identifying R∗+ × H with GL+
2 (R)/SO2 (R) we conclude that the flow of weights of
the factor πϕβ (A)00 is defined on the quotient of the measure space (Mat2 (A), λβ ) by the action of
GL2 (Q) × SO2 (R) defined by (g, s)(m, ρ) = (gms, gρ) for (g, s) ∈ GL2 (Q) × SO2 (R) and (m, ρ) ∈
Mat2 (A) = Mat2 (R) × Mat2 (Af ).
It seems natural to conjecture that the action of GL2 (Q) on (Mat2 (A), λβ ) is ergodic, so the flows
of weights of the factors πϕβ (A)00 and πϕ̄β (B)00 are trivial, and thus the factors are of type III1 .
The analogous property in the one-dimensional case indeed holds [3, 18]. Note that so far we have
only shown that the action of GL2 (Q) × R∗ is ergodic, which is equivalent to ergodicity of the
+
action of GL+
2 (Q) on (PGL2 (R) × Mat2 (Af ), µ̄β ). Note also that similarly to the one-dimensional
PHASE TRANSITION IN THE CONNES-MARCOLLI GL2 -SYSTEM
23
case [18], by virtue of Lemma 4.5 and Proposition 4.6, to prove the conjecture it would be enough
+
to show that the action of GL+
2 (Q) on GL2 (R) × (Mat2 (Af )/GL2 (Ẑ)) is ergodic, or equivalently, the
action of GL2 (Q) on (Mat2 (A)/GL2 (Ẑ), λβ ) is ergodic. Recall that in the one-dimensional case the
corresponding ergodicity result for the action of Q∗ on A/Ẑ∗ was established in [1] and [2].
Remark 4.10. We believe that the results of Sections 3 and 4 are valid for GLn for any n ≥ 2.
+
More precisely, consider the algebra Cr∗ (SLn (Z)\GL+
n (Q) SLn (Z) (PGLn (R) × Matn (Ẑ))). Define a
+
dynamics by the homomorphism GLn (Q) 3 g 7→ det(g). Then
(i) for β ∈ (−∞, 0) ∪ (0, 1) ∪ · · · ∪ (n − 2, n − 1) there are no KMSβ -states;
(ii) for β ∈ (n − 1, n] there exists a unique KMSβ -state;
(iii) for β > n there is a one-to-one correspondence between KMSβ -states and probability measures
on SLn (Z)\(PGL+
n (R) × GLn (Ẑ));
(iv) for β = 0, 1, . . . , n − 1 there is a KMSβ -state defined by the Haar measure on Aβn
f , when we
identify the latter group with the set of matrices in Matn (Af ) with zero first n − β columns.
The key step for this generalization would be an analogue of Lemma 3.5.
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Department of Mathematics and Statistics, University of Victoria, PO Box 3045, Victoria, British
Columbia, V8W 3P4, Canada.
E-mail address: laca@math.uvic.ca
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.
E-mail address: nadiasl@math.uio.no
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.
E-mail address: sergeyn@math.uio.no